Exact solutions for free vibration analysis of laminated, box and sandwich beams by refined layer-wise theory Yan Yang a,b , Alfonso Pagani b , Erasmo Carrera b,⇑ a College of Mechanics and Materials, Hohai University, 210098 Nanjing, China b Mul 2 , Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy article info Article history: Received 15 March 2017 Revised 25 April 2017 Accepted 2 May 2017 Available online 5 May 2017 Keywords: Carrera unified formulation Layer-wise approach Closed-form solution Free vibration analysis abstract The present work addresses a closed-form solution for the free vibration analysis of simply supported composite laminated beams via a refined one-dimensional (1D) model, which employs the Carrera Unified Formulation (CUF). In the framework of CUF, the 3D displacement field can be expanded as any order of generic unknown variables over the cross section, in the case of beam theories. Particularly, Lagrange expansions of cross-sectional displacement variables in conjunction with layer- wise (LW) theory are adopted in this analysis, which makes it possible to refine the kinematic fields of complex cross section by arbitrary order and accuracy. As a consequence, the governing equations can be derived using the principle of virtual work in a unified form and can be solved by a Navier-type, closed-form solution. Numerical investigations are carried out to test the performance of this novel method, including composite and sandwich beams ranging from simple to complex configurations of the cross section. The results are compared with those available in the literature as well as the 3D finite element method (FEM) solutions computed by commercial codes. The present CUF model is proved to be able of achieving high accurate results with less computational costs. Besides, they may serve as bench- marks for future assessments in this field. Ó 2017 Elsevier Ltd. All rights reserved. 1. Introduction Composite beams, as basic structural components, have been widely used in various engineering fields such as aerospace, mechanical, civil and ocean engineering due to their high strength- and stiffness-to-weight ratios. Also, determination of vibration characteristics is of crucial importance in the safe design of composite beams subjected to dynamic loads. Compared with the isotropic homogeneous elastic beam [1], composite structures present more complex material properties (anisotropy as well as fiber angle, and laminate stacking sequence), resulting in non clas- sical vibration modes phenomena with couplings between torsion, shear and bending. These effects cannot be detected by 1D lower- order models, which were firstly extrapolated from classical theo- ries under the assumptions outlined by Euler–Bernoulli [2]. As a result, it becomes essential to develop a simple yet accurate com- posite beam model to describe these specific mechanical beha- viours correctly. Refined 1D beam models have received widespread attention owing to their simplicity and higher-efficient computing perfor- mance. Over the years, several 1D refined composite beam models have been systematically developed for different engineering pur- poses. As far as the free vibration analysis is concerned, a brief overview of recent research on these refined 1D models is reported here. The first-order shear deformation theory (FSDT), as the improvement of Euler–Bernoulli beam theory, was proposed as an extension of the plate theories of Reissner [3] and Mindlin [4], which assume a constant transverse shear deformation in the thickness direction. Nevertheless, this assumption does not con- form to stress-free boundary conditions. Thus, a shear correction factor was introduced to correct this theory and contributed to fruitful results [5,6]. Since accurate estimation of the shear correc- tion factor exerts much effort, several high-order shear deforma- tion theories (HSDT) were proposed, which provided different distributions of the transverse shear strains along the thickness. In details, Khedeir and Reddy [7], employed a parabolic form of HSDT to study the free vibration behaviour of cross-ply laminated beams with arbitrary boundary conditions via a Navier-type ana- lytical solution. Arya et al. [8] presented a trigonometric HSDT for the static analysis of symmetric cross-ply laminated beam, http://dx.doi.org/10.1016/j.compstruct.2017.05.003 0263-8223/Ó 2017 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: [email protected](Y. Yang), [email protected](A. Pagani), [email protected](E. Carrera). Composite Structures 175 (2017) 28–45 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct
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Yan Yang a,b, Alfonso Pagani b, Erasmo Carrera b,⇑aCollege of Mechanics and Materials, Hohai University, 210098 Nanjing, ChinabMul2, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
a r t i c l e i n f o
Article history:Received 15 March 2017Revised 25 April 2017Accepted 2 May 2017Available online 5 May 2017
The present work addresses a closed-form solution for the free vibration analysis of simply supportedcomposite laminated beams via a refined one-dimensional (1D) model, which employs the CarreraUnified Formulation (CUF). In the framework of CUF, the 3D displacement field can be expanded asany order of generic unknown variables over the cross section, in the case of beam theories.Particularly, Lagrange expansions of cross-sectional displacement variables in conjunction with layer-wise (LW) theory are adopted in this analysis, which makes it possible to refine the kinematic fields ofcomplex cross section by arbitrary order and accuracy. As a consequence, the governing equations canbe derived using the principle of virtual work in a unified form and can be solved by a Navier-type,closed-form solution. Numerical investigations are carried out to test the performance of this novelmethod, including composite and sandwich beams ranging from simple to complex configurations ofthe cross section. The results are compared with those available in the literature as well as the 3D finiteelement method (FEM) solutions computed by commercial codes. The present CUF model is proved to beable of achieving high accurate results with less computational costs. Besides, they may serve as bench-marks for future assessments in this field.
� 2017 Elsevier Ltd. All rights reserved.
1. Introduction
Composite beams, as basic structural components, have beenwidely used in various engineering fields such as aerospace,mechanical, civil and ocean engineering due to their highstrength- and stiffness-to-weight ratios. Also, determination ofvibration characteristics is of crucial importance in the safe designof composite beams subjected to dynamic loads. Compared withthe isotropic homogeneous elastic beam [1], composite structurespresent more complex material properties (anisotropy as well asfiber angle, and laminate stacking sequence), resulting in non clas-sical vibration modes phenomena with couplings between torsion,shear and bending. These effects cannot be detected by 1D lower-order models, which were firstly extrapolated from classical theo-ries under the assumptions outlined by Euler–Bernoulli [2]. As aresult, it becomes essential to develop a simple yet accurate com-posite beam model to describe these specific mechanical beha-viours correctly.
Refined 1D beam models have received widespread attentionowing to their simplicity and higher-efficient computing perfor-mance. Over the years, several 1D refined composite beam modelshave been systematically developed for different engineering pur-poses. As far as the free vibration analysis is concerned, a briefoverview of recent research on these refined 1D models is reportedhere. The first-order shear deformation theory (FSDT), as theimprovement of Euler–Bernoulli beam theory, was proposed asan extension of the plate theories of Reissner [3] and Mindlin [4],which assume a constant transverse shear deformation in thethickness direction. Nevertheless, this assumption does not con-form to stress-free boundary conditions. Thus, a shear correctionfactor was introduced to correct this theory and contributed tofruitful results [5,6]. Since accurate estimation of the shear correc-tion factor exerts much effort, several high-order shear deforma-tion theories (HSDT) were proposed, which provided differentdistributions of the transverse shear strains along the thickness.In details, Khedeir and Reddy [7], employed a parabolic form ofHSDT to study the free vibration behaviour of cross-ply laminatedbeams with arbitrary boundary conditions via a Navier-type ana-lytical solution. Arya et al. [8] presented a trigonometric HSDTfor the static analysis of symmetric cross-ply laminated beam,
Fig. 1. Physical and material coordinate systems for a laminated composite beam.
Y. Yang et al. / Composite Structures 175 (2017) 28–45 29
and Li et al. [9] extended this refined model to study free vibrationof angle-ply laminated beam with general boundary conditions.Vidal and Polit [10] introduced a three-node beam element to per-form the free vibration of composite and sandwich beams based onthe trigonometric HSDT. A exponential HSDT was used for thebending, buckling and free vibration analyses of multi-layeredlaminated composite beams by Karama et al. [11], showing thatthe proposed model was more precise than the trigonometricHSDT model and FEM model studied early by Karama et al. [12].In addition, other HSDT models [13] have been developed by vari-ous authors for describing the deformation through the thickness.
It should be noted that the above models were implemented onthe basis of an Equivalent Single Layer (ESL) approach, whichhypothesizes a continuous and differentiable displacement func-tion through the thickness direction. Unfortunately, this assump-tion cannot account for the continuity of the transverse stressesand the zig–zag behavior of the displacements along the thickness.Therefore, a more precise hypothesis called layer-wise theory wasput forward to overcome this drawback. In the domain of LW, acontinuous displacement function is adopted for each layer, and,as a consequence, a discontinuous derivative of displacement func-tion is imposed at the intra-layer interfaces, thereby, meeting thefundamental requirements demanded by modelling of laminatedstructures. Shimpi and Ainapure [14] used LW theory to studythe natural frequencies of simply supported two-layer beam incombination with the trigonometric HSDT. Tahani [15] investi-gated the static and dynamic properties of composite beam withgeneral laminations using two different strategies based on LWtheory. Plagianakos and Saravanos [16] applied the finite elementmethod to predict damping and natural frequencies of thick com-posite and sandwich beams via a parabolic HSDT in conjunctionwith LW theory.
In contrast to ESL theory, burdensome computation cost may berequired in LW theory, being dependent on the number of laminatelayers. Therefore, several layer-independent theories have beendeveloped on the premise of additional computational capacityand consumed time. In these theories, zig–zag or Heaviside func-tions were added in the framework of ESL theory. Carrera [17] pre-sented a thorough review of Murakami’s zig–zag method [18], whoadded a zig–zag function to approximate the thickness distributionof in-plane displacements. Furthermore, Carrera et al. [19]extended this theory to the static analysis of symmetric andantisymmetric cross-ply laminated beams, based on polynomial,trigonometric, exponential HSDT, respectively. Filippi and Carrera[20] made use of a higher-order zig–zag function to predict thenatural frequencies of laminated and sandwich beams with lowerslenderness ratio values. Other classes of Heaviside functions canbe found in [21,22].
Although the above refined theories can improve the accuracyof results significantly, it is a matter of fact that many of themare problem-dependent. Motivated by this deficiency, it is of nota-ble importance to introduce a unified formulation which can besuitable for any structural composite beam. Carrera et al. [23] pro-posed this unified formulation, which was later denoted to as Car-rera Unified Formulation (CUF). CUF was originally considered forthe analysis of plate and shell structures, hereafter referred to as2D CUF [24–26] and continued to be employed for beam struc-tures, hereafter referred to as 1D CUF [27,28]. In the light of 1DCUF, the 3D displacement field can be expanded elegantly as anyorder of the generalized unknown variables over the cross section.Moreover, the order of expansions can be regarded as a free param-eter depending on the problem under consideration. In addition,FSDT and HSDT can be effortlessly derived in a hierarchical andcompact way in the domain of CUF. Carrera et al. [29] used Taylorseries polynomials as displacement expansions to obtain 3D stressstates of beams with arbitrary cross-sectional geometries via 1D
CUF FEM. The corresponding model is called 1D CUF Taylor expan-sion (TE), which has been likewise applied to the free vibrationanalysis [30–32], buckling phenomenon [33], composite beams[34–36] and functional graded beams [37,38]. Recently, a new classof CUF model was proposed by Carrera and Petrolo. [39], wheredisplacements are approximated by the sum of cross-sectionalnode displacement unknowns via Lagrange expansion (LE), beinginhere LW ability. This 1D CUF LE model permits one to analyzebehaviours of beams with more complex geometry shape with lesscomputational costs. Carrera et al. [40] adopted TE and LE CUF forthe in-plane and out-of-plane stress analysis of compact and multi-cell laminated box beams by using FEM, in which, TE is imple-mented along with ESL, whereas, LW is carried out in the frame-work of LE. Then, the same authors extended static analysis tothe free vibration problem, readily detecting solid and shell-likephenomena [41]. Other important CUF model are those on thebasis of Chebyshev Expansions (CE) [42] and HierarchicalLegendre-type Expansions (HLE) [43].
In contrast to the 1D CUF model solved by weak-form solution,e.g., FEM, Giunta et al. provided a strong-form solution, namely,Navier-type solution of the 1D CUF TE governing equation for thefree vibration analysis of composite beam [37] and staic, bucklingand free vibration analysis of sandwich beams [44,45]. The exten-sion of the Navier-type closed-form solution to the 1D CUF LE forfree vibration analysis of isotropic beams was done by Dan et al.[46].
In the present paper, for the first time, the same analytical solu-tion is utilized for the free vibration of cross-ply composite beamwith compact and thin-walled cross sections subject to the simplysupported boundary conditions based on 1D CUF LE model and LWtheory. The rest of this paper is structured as follows: (i) a briefintroduction of anisotropic elasticity theory and 1D CUF LE theoryare given in Section 2.1; (ii) the equations of motion and corre-sponding boundary conditions are derived using principle of vir-tual work in Section 2.2 and a linear eigensystem is obtainedusing the Navier-type closed-form solution in Section 2.3; (iii)the numerical results of different assessments considered are pre-sented in Section 3; (iv) some conclusions and remarks of this workare outlined in the last section.
Fig. 2. Cross sections for two- and three-layer laminated beams.
Table 1First four non-dimensional natural frequencies x� for a two-layer composite beam [0/90] with m = 1, L=b ¼ 100.
a Flexural mode on plane yz.b Flexural (plane xy)/torsional mode.c Torsional mode.d Axial/shear (plane yz) mode.e Mode not provided by the theory.f The number of elements is 8� 80� 8.g The number of elements is 6� 60� 6.
Table 2First five non-dimensional natural frequencies x� for a two-layer composite beam [0/90] with m = 1, L=b ¼ 5
Model DOFs mode 1a mode 2b mode 3c mode 4d mode 5e
FEM 3D20g [35] 1037043 4.9357 6.4491 9.0672 33.566 50.448
FEM 3D6h [35] 33159 4.9387 6.4520 9.0698 33.564 50.441
a Flexural mode on plane yz.b Flexural (plane xy)/torsional mode.c Torsional mode.d Axial/shear (plane yz) mode.e Shear mode on plane xz.f Mode not provided by the theory.g The number of elements is 20� 200� 20.h The number of elements is 6� 60� 6.
30 Y. Yang et al. / Composite Structures 175 (2017) 28–45
Table 3First four non-dimensional natural frequencies x� for a three-layer composite beam [0/90/0] with m = 1, l=b ¼ 100.
a Flexural mode on plane xy.b Flexural mode on plane yz.c Torsional mode.d Axial mode.e Mode not provided by the theory.f The number of elements is 9� 90� 9.g The number of elements is 6� 60� 6.
Table 4First five non-dimensional natural frequencies x� for a three-layer composite beam [0/90/0] with m = 1, l=b ¼ 5.
Model DOFs mode 1a mode 2b mode 3c mode 4d mode 5e
FEM 3D24g [35] 1769475 6.8888 7.4965 9.0386 55.536 57.912
FEM 3D12h [35] 235443 6.8894 7.4972 9.0391 55.536 57.913
a Flexural mode on plane yzb Flexural mode on plane xyc Torsional moded Shear mode on plane xze Axial/shear (plane yz) modef Mode not provided by the theoryg The number of elements is 24� 240� 24.h The number of elements is 12� 120� 12.
Y. Yang et al. / Composite Structures 175 (2017) 28–45 31
2. 1D CUF beam theory
2.1. Preliminaries
Consider a multi-layer laminated beam in physical coordinatesystem, as shown in Fig. 1. Assume that y-axis is coincident withthe longitudinal axis of the beam and its cross section is definedon the xz-plane, being denoted as X. The superscript k stands forthe number of the generic layer, starting from the bottom to top.The three-dimensional displacement vector for kth layer is intro-duced as follows:
ukðx; y; z; tÞ ¼ ukx uk
y ukz
n oTð1Þ
where ukx ;u
ky and uk
z indicate the displacement components alongthree axes x; y; z, respectively. The index ‘‘T” denotes the transposeoperator. Similarly, stress r and strain � components can bearranged as:
rk ¼ rkyy r
kxx r
kzz r
kxz r
kyz r
kxy
n oT; �k
¼ �kyy �kxx �
kzz �
kxz �
kyz �
kxy
n oTð2Þ
Based on the assumption of small displacements and strains,the relationship between ðrÞ and ð�Þ can be expressed as:
�k ¼ Duk ð3Þwhere
Fig. 3. Selected mode shapes of two- and three-layer laminated beams of Table 1 and Table 2 via the 2� 2L16 and 3� 3L16 model, m = 1.
Fig. 4. Cross sections for nine- and ten-layer laminated beams.
32 Y. Yang et al. / Composite Structures 175 (2017) 28–45
Table 5First five non-dimensional natural frequencies x� for a ten-layer anti-symmetric cross-ply laminated composite beam with m = 1–5, l=b ¼ 5.
Y. Yang et al. / Composite Structures 175 (2017) 28–45 33
Fig. 5. The lowest mode shapes 1–8 for an anti-symmetric cross-ply ten-layer laminated composite beam (l=b ¼ 5) of Table 5 via the 5� 10L16 model, with m = 1–3.
34 Y. Yang et al. / Composite Structures 175 (2017) 28–45
D ¼
0 @@y 0
@@x 0 00 0 @
@z@@z 0 @
@x
0 @@z
@@y
@@y
@@x 0
26666666664
37777777775
ð4Þ
In the case of laminated composite materials, the constitutiveequation for kth layer assumes the following form:
rk ¼ eCk�k ð5Þwhere
eCk ¼
eCk11
eCk12
eCk13 0 0 eCk
36eCk21
eCk22
eCk23 0 0 eCk
26eCk31
eCk32
eCk33 0 0 eCk
16
0 0 0 eCk44
eCk45 0
0 0 0 eCk45
eCk55 0eCk
16eCk26
eCk36 0 0 eCk
66
266666666664
377777777775
ð6Þ
Coefficients in the matrix above are function of three parame-ters: Young modulus, Poisson ratios and fiber orientation angle(h) measured down from the positive y-axis. For the sake of brevityand clarity, we do not provide the detailed expressions, one canrefer to Reddy [47] for further details.
The generic displacement field, within the framework of CUF,can be expanded as arbitrary functions Fs:
where Fs is a function depending on the x and z coordinates. us isthe generic displacements vector of axial coordinates y. M is thenumber of expanded terms, and the repeated subscript, s, standsfor summation.
In this study, Lagrange expansion polynomials are employed asthe function Fs to discrete the arbitrarily complex cross section,whose approximation precision lies on the order of LE polynomials.Three types of LE polynomials, i.e., four-node quadrilateral L4,nine-node cubic L9, and sixteen-node quartic L16 polynomials,are often adopted. The expression of L9 polynomial is presentedhere as an illustrative example:
Fs ¼ 14ðr2 þ r rsÞðs2 þ s ssÞ s ¼ 1;3;5;7
Fs ¼ 12s2sðs2 � s ssÞð1� r2Þ þ 1
2r2sðr2 � r rsÞð1� s2Þ s ¼ 2;4;6;8
Fs ¼ ð1� r2Þð1� s2Þ s ¼ 9
ð8Þ
where r and s vary over the interval [�1;þ1], and rs and ss indicatethe vertex location in the natural coordinate system. For moredetails about the other two kinds of LE polynomials, see Carreraand Petrolo [39].
Fig. 6. The lowest mode shapes 1–8 for a symmetric nine-layer cross-ply laminated composite beam (l=b ¼ 5) of Table 6 via the 5� 9L16 model, with m = 1–3.
Y. Yang et al. / Composite Structures 175 (2017) 28–45 35
The nine-node cubic single-L9 kinematic field is therefore givenby:
ukx ¼F1uk
x1þF2uk
x2þF3uk
x3þF4uk
x4þF5uk
x5þF6uk
x6þF7uk
x7þF8uk
x8þF9uk
x9
uky¼F1uk
y1þF2uk
y2þF3uk
y3þF4uk
y4þF5uk
y5þF6uk
y6þF7uk
y7þF8uk
y8þF9uk
y9
ukz ¼F1uk
z1þF2uk
z2þF3uk
z3þF4uk
z4þF5uk
z5þF6uk
z6þF7uk
z7þF8uk
z8þF9uk
z9
ð9Þ
where ukx1; . . . ; uk
z9are the nine-node translational displacement
variables of the problem considered.
The present LE model can be refined either with higher-orderpolynomials (global refinement) or a combination of polynomialsin each sub-domain cross section (local refinement). For the sakeof brevity, the following derivations are carried out on the kth layerand superscript k will be omitted.
2.2. Equations of motion
Equations of motion and corresponding boundary conditionscan be obtained via the variational principle of virtual work.
dL ¼ dLint þ dLine ¼ 0 ð10Þwhere d is the symbol of a virtual variation. Lint stands for the strainenergy, Line represents the inertial work.
Table 8First five non-dimensional natural frequencies x� for a three-layer sandwich beam [0/0/0] with m = 1–5, l=b ¼ 5.
a The number of elements is 10� 50� 10 using reduced integration.b The number of elements is 10� 50� 10 using full integration.
36 Y. Yang et al. / Composite Structures 175 (2017) 28–45
The strain energy can be expressed as follows:
dLint ¼ZVd�TrdV ð11Þ
Substituting Eq. (3), Eq. (5) and Eq. (7) into Eq. (11) and usingthe integration by parts (see [48]), one has:
dLint ¼ZlðdusÞTKssusdyþ ½ðdusÞTPssus�jy¼l
y¼0 ð12Þ
where Kss is fundamental nucleus of the stiffness, Pss representsthe mechanical boundary conditions and l is the length of the beam.They are both 3� 3 matrices. For the sake of brevity, the explicitexpressions concerning these fundamental nuclei are not reportedhere, but are available from the corresponding literature [36]. It is
envisaged that the term ½ðdusÞTPssus�jy¼ly¼0 is equal to zero in the case
of simply supported beam and will be removed in the followingequations.
The virtual variation of the inertial work is defined as:
dLine ¼ZVqdu€udV ð13Þ
where q stands for the density of material and the superimposeddots denote double derivative with respect to time (t). Accountingfor Eq. (7), Eq. (13) can be rewritten as:
dLine ¼ZldusM
ss€usdy ð14Þ
The components of the 3� 3 mass matrix Mss are:
Mssij ¼ dijE
qss i; j ¼ 1; . . . ;3 ð15Þ
where dij is the Dirac’s delta function and:
Eqss ¼ZXqFsFsdX ð16Þ
The explicit expression of the dynamic governing equations canbe obtained from the principle of virtual displacements as follows:
duxs :� E66ss uxs;yy þ E26
s;xs � E26ss;x
� �uxs;y þ E22
s;xs;x þ E44s;zs;z
� �uxs
� E36ss uys;yy þ E23
s;xs � E66ss;x
� �uys;y þ E26
s;xs;x þ E45s;zs;z
� �uys
þ E45s;zs � E16
ss;z
� �uzs;y þ E44
s;zs;x þ E12s;xs;z
� �uzs ¼ �Eqss€uxs
duys :� E36ss uxs;yy þ E66
s;xs � E23ss;x
� �uxs;y þ E26
s;xs;x þ E45s;zs;z
� �uxs
� E33ss uys;yy þ E36
s;xs � E36ss;x
� �uys;y þ E66
s;xs;x þ E55s;zs;z
� �uys
þ E55s;zs � E13
ss;z
� �uzs;y þ E16
s;xs;z þ E45s;zs;x
� �uzs ¼ �Eqss€uys
duzs : E16s;zs � E45
ss;z
� �uxs;y þ E44
s;xs;z þ E12s;zs;x
� �uxs
þ E13s;zs � E55
ss;z
� �uys;y þ E45
s;xs;z þ E16s;zs;x
� �uys � E55
ss uzs;yy
þ E45s;xs � E45
ss;x
� �uzs;y þ E44
s;xs;x þ E11s;zs;z
� �uzs ¼ �Eqss€uzs
ð17Þ
Fig. 8. The core modes from the top view for a three-layer sandwich beam of Table 8 via the 6� 9L16 model, with m = 1–5.
Y. Yang et al. / Composite Structures 175 (2017) 28–45 37
where the suffix after the comma indicates the derivatives and thegeneric term Eabs;hs;f is a cross-sectional moment parameter:
Eabs;hs;f ¼ZX
eCabFs;hFs;fdX ð18Þ
2.3. Analytical solution
In the case of simply supported composite beam, the analyticalsolution of the above differential equations can be obtained via aNavier-type solution. The displacement fields are assumed as asum of harmonic functions:
Fig. 9. The cross section for a T-shaped composite beam.
38 Y. Yang et al. / Composite Structures 175 (2017) 28–45
uxsðy; tÞ ¼ Uxs sinðayÞeixt
uysðy; tÞ ¼ Uys cosðayÞeixt ð19Þ
uzsðy; tÞ ¼ Uzs sinðayÞeixt
where a is:
a ¼ mpl
ð20Þ
where Uxs;Uys and Uzs are the amplitudes of the components of thegeneralized displacements vector. m is the half wave number alongthe beam axis, x is the vibrational natural frequency and i is theimaginary unit. Substituting Eq. (19) into Eq. (17), it holds:
Table 9First five natural frequencies (Hz) for a T-shaped composite beam with m = 1–5, l=b ¼ 10
Cross section Non-dimensiona
Seq. Model DOFs Mode:1
43L4 234 68.889m ¼ 1 29L9 483 68.508
29L16 984 68.362FEM 3Da 126765 68.379
32L4 209.80m ¼ 2 29L9 208.69
29L16 208.29FEM 3Da 208.31
32L4 436.35m ¼ 3 29L9 434.12
29L16 432.90FEM 3Da 432.98
32L4 747.05m ¼ 4 29L9 743.17
29L16 739.90FEM 3Da 740.12
32L4 1134.4m ¼ 5 29L9 1126.8
29L16 1119.5FEM 3Da 1120.1
The number of elements in each web is 2� 50� 50.a The number of elements in each flange is 25� 50� 2.
dUxs :a2E66ss Uxs sinðayÞ þ a E26
s;xs � E26ss;x
� �Uxs cosðayÞ
þ E22s;xs;x þ E44
s;zs;z
� �Uxs sinðayÞ
þ a2E36ss Uys cosðayÞ � a E23
s;xs � E66ss;x
� �Uys sinðayÞ
þ E26s;xs;x þ E45
s;zs;z
� �Uys cosðayÞ
þ a E45s;zs � E16
ss;z
� �Uzs cosðayÞ þ E44
s;zs;x þ E12s;xs;z
� �Uzs sinðayÞ
¼ x2EqssUxs sinðayÞdUys :a2E36
ss Uxs sinðayÞ þ a E66s;xs � E23
ss;x
� �Uxs cosðayÞ
þ E26s;xs;x þ E45
s;zs;z
� �Uxs sinðayÞ
þ a2E33ss Uys cosðayÞ � a E36
s;xs � E36ss;x
� �Uys sinðayÞ
þ E66s;xs;x þ E55
s;zs;z
� �Uys cosðayÞ
þ a E55s;zs � E13
ss;z
� �Uzs cosðayÞ þ E16
s;xs;z þ E45s;zs;x
� �Uzs sinðayÞ
¼ x2EqssUys cosðayÞdUzs :a E16
s;zs � E45ss;z
� �Uxs cosðayÞ þ E44
s;xs;z þ E12s;zs;x
� �Uxs sinðayÞ
� a E13s;zs � E55
ss;z
� �Uys sinðayÞ þ E45
s;xs;z þ E16s;zs;x
� �Uys cosðayÞ
þ a2E55ss Uzs sinðayÞ
þ a E45s;xs � E45
ss;x
� �Uzs cosðayÞ þ E44
s;xs;x þ E11s;zs;z
� �Uzs sinðayÞ
¼ x2EqssUzs sinðayÞð21Þ
It is important to underline that the governing equation can be
decoupled by setting the material parameters eC16; eC26; eC36; eC45 tobe zero, which means isotropic or cross-ply laminate beams. Thusthe above equations can be converted into the algebraic eigensys-tem as:
Fig. 10. The lowest mode shapes 1–9 for a T-shaped laminated composite beam of Table 9 via the 29L16 model, with m = 1–4.
Fig. 11. The cross section for a single-bay composite box beam.
Y. Yang et al. / Composite Structures 175 (2017) 28–45 39
Kssxx¼a2E66
ss þE22s;xs;x þE44
s;zs;z ;Kssxy¼a E23
s;xs�E66ss;x
� �;Kss
xz ¼E44s;zs;x þE12
s;xs;z
Kssyx¼a E66
s;xs�E23ss;x
� �;Kss
yy¼a2E33ss þE66
s;xs;x þE55s;zs;z ;K
ssyz¼a E55
s;zs�E13ss;z
� �
Ksszx¼E44
s;xs;z þE12s;zs;x ;K
sszy¼a2E55
ss þa E13s;zs�E55
ss;z
� �;Kss
zz ¼E44s;xs;x þE11
s;zs;z
Mssxx¼Mss
yy¼Msszz ¼Eqss;M
ssxy¼Mss
xz ¼Mssyx¼Mss
yz¼Msszx ¼Mss
zy¼0
ð23Þ
The corresponding mechanical and natural boundary conditionscan be also simplified as follows:
Uxs ¼ 0Uys;y ¼ 0
Uzs ¼ 0
ð24Þ
Eq. (22) is assessed for kth layer and can be assembled into aglobal algebraic eigensystem in the light of contribution of eachlayer. Layer-wise theory is used to fulfill this procedure, whichcan be referred to Pagani et al. [43] for the sake of simplicity. In thispaper, LW models are implemented by utilizing one or more LEexpansions on the cross-sectional domain of each layer, as dis-cussed in the following sections. As a consequence, the theorykinematics can be opportunely varied at layer level by settingthe order of LE expansions. This characteristic of LE CUF beammodels allows the implementation of higher-order LW models inan easy and straightforward manner.
3. Numerical results
To demonstrate the exactness of the proposed quasi-3D modelon the basis of CUF, free vibration analysis of simply supportedcomposite beams with solid and thin-walled cross sections areinvestigated. The first part of this section focuses on the compactsquare cross-ply laminated beams considering different slender-ness ratios, number of layers for laminates and material propertiesbetween layers, while the second part of this section presents thin-
Table 10First five natural frequencies (Hz) for a single-bay composite box beam with m = 1–5, l=b ¼ 10.
3.1.1. Two- and three-layer laminated beamsSquare-sectional beams, consisting of two-layer [0/90] and
three-layer [0/90/0] laminates of the same thickness, are consid-ered in the first assessment. The dimensions of the beam are ofequal width and height: b ¼ h ¼ 0:2 m, being two kinds of slender-ness ratios: l=b ¼ 100 (slender beam) and l=b ¼ 5 (short beam). Thematerial is assumed to be orthotropic with the following proper-ties: Young modulus: EL ¼ 250 GPa, ET ¼ 10 GPa; Poisson ratio:mLT ¼ mTT ¼ 0:33; Material density: q ¼ 2700 kg=m3; Shear modu-lus: GLT ¼ 5 GPa, GTT ¼ 2 GPa, where the subscripts L and T repre-sent the direction parallel and perpendicular to the fibres,respectively.
Unless differently specified, we use the notation f� gLb todenote beams of square cross sections, whereas #Lb to denotethose of thin-walled cross sections, where f and g stand for thenumber of Lb elements in the x direction and z direction, # standsfor the number of Lb elements over the whole cross section, and bstands for bilinear(4), cubic(9) and fourth-order(16) Lagrange poly-nomials, respectively.
Fig. 2a and Fig. 2b present the cross sections of the laminatedbeams addressed.
Tables 1 and 3 show a list of the first four non-dimensional nat-ural frequencies with one half wave number (m ¼ 1) for slender(l=b ¼ 100) two- and three-layer composite beams. Moreover, inTables 2 and 4 the first five non-dimensional natural frequencieswith one half wave number (m ¼ 1) for short (l=b ¼ 5) two- andthree-layer composite beams are given. The number of Degreesof Freedoms (DOFs) for different models are also reported in thesecond column of the tables. The results obtained by various LEmodels are compared with the classical beam models, including
Euler–Bernoulli beam model (EBBM) and Timoshenko beam model(TBM), and refined closed-form CUF-TE solutions provided byGiunta et al. [35]. Three-dimensional finite element model createdby Ansys software also serves as a benchmark for the same assess-ment, where the quadratic solid element SOLID 186 is used. Twodifferent mesh schemes (coarse mesh and refined mesh) areadopted to ensure the convergence, and the notation FEM 3Dn
denotes the solid model with n elements along the x-axis, n� 10elements along y-axis, and n elements along z. The results are givenin terms of the following non-dimensional natural frequency x�:
From Table 1, it can be seen that the present CUF-LE theory witheven the simplest elements (1� 2L4 and 2� 2L4) shows the sameaccuracy as EBBM and TBM. On the other hand, the resultsobtained by the higher-order LW proposed models achieve fasterconvergence to the refined FEM 3D g than refined CUF-TE theories[35].
In the case of short beams, EBBm is proved to be incapable ofobtaining the correct result and so has TBM (but mode 1), whichis shown in Table 2. In addition, present lower-order CUF-LE mod-els (1� 2L4 and 2� 2L4) and refined lower-order CUF-TE model(N ¼ 2) yield poor results in mode 4 and mode 5, i.e. in the caseof axial and shear modes. Conversely, higher-order models makinguse of L9 and L16 LW approximation can produce the same resultsas 3D FEM solutions with less computational costs.
From Tables 3 and 4, it is obvious that the majority of modes aresymmetric modes without coupling effects except for mode 5 inthe case of l=b ¼ 5. Meanwhile, it is noteworthy that EBBM andTBM are considered not sufficient for predicting the first twomodes in Table 4. Moreover, more attention should be paid tomodel 3� 3L9 and 1� 3L16, which produce approximately thesame solutions independently from the number of DOFs. Thisimportant observation implies that CUF-LE model with higherorder expansion is able to detect each mode exactly regardless of
Fig. 12. Comparison of the fifth mode shapes for m = 2–5, by 32L16 and 3D FEM model.
Y. Yang et al. / Composite Structures 175 (2017) 28–45 41
the slenderness ratio though lacking enough elements in both xand z directions.
Three selected mode shapes, i.e. mode 1, mode 2, and mode 4,concerning two- and three-layer composite beams (l=b ¼ 5) and
obtained by 2� 2L16 and 3� 3L16 models, are shown in Fig. 3.From these graphs, it should be underlined that coupled flexural/-torsion and axial/shear phenomena appear when unsymmetriclamination is considered. Beyond that, shear mode on plane yz is
Fig. 13. The cross section for a composite sandwich beam
42 Y. Yang et al. / Composite Structures 175 (2017) 28–45
apt to appear in mode 4 for the two-layer case, while mode 4 isdominated by shear mode on plane xz in the other case.
3.1.2. Nine- and ten-layer laminated beamsThis section aims to investigate the vibration characteristics of
composite beam constructed by nine and ten layers. A ten-layeranti-symmetric and nine-layer symmetric cross-ply laminatedbeams are separately considered (see Fig. 4a and b). Tables 5 and6 present the first five corresponding non-dimensional natural fre-quencies with m ¼ 1 to m ¼ 5 via the current model and 3D FEMsoftware ABAQUS. From Table 5, it is possible to see that the pre-sent 5� 10L16 model predicts lower values than 3D FEM model
Table 11First five natural frequencies (Hz) for a composite sandwich-box beam with m = 1–5, l=b ¼
in most frequencies, which suggests that higher-order LE modelwith thousands of DOFs overcomes the results provided by 3DFEM model with one hundred thousands of DOFs. Again,1� 10L16 model can produce almost the same value in compar-ison with 5� 10L9 with nearly half of its DOFs. Besides, as forhigher number of half-waves (m ¼ 2 and m ¼ 4) the lower-ordermodel (2� 10L4) interchanges the order of appearance for the fol-lowing two cases (see mode 5 in m ¼ 2 and mode 4 in m ¼ 4). Thesame conclusion, i.e., the high efficiency in higher-order model andmodal confusion in lower-order model, can be also observed inTable 6. Comparing Table 6 and Table 5, it is worth mentioning thateach modes remain almost the same for both of cases, regardless ofthe value of m. Figs. 5 and 6 display the lowest mode shapes 1–8corresponding to anti-symmetric and symmetric cross-ply lami-nated composite beams (l=b ¼ 5) via 5� 10L16 model and5� 9L16 model, with m = 1–5. Out of these figures, it is importantto underline that for higher m values (m ¼ 2 and m ¼ 3), torsionmode tends to appear before the dominant flexural mode on planexy in both of cases.
3.2. Sandwich beam
A three-layer sandwich beam with a soft core is further consid-ered (see Fig. 7). The geometric parameters of the beam are as fol-lows: b ¼ h ¼ 0:2, length-to-width ratio l=b ¼ 5. The thicknesses oftop face and bottom face are equal: hf ¼ hb ¼ 0:02 m, whereas thethickness of the core is hc ¼ 0:16 m. The material properties aregiven in Table 7. The first five non-dimensional natural frequencieswith m = 1–5, computed by the present model and 3D FEM solu-tions, are reported in Table 8. From this table, it can be seen thatasm increases, the gap between each mode (Modes 1–5) decreases,especially in the case m ¼ 5, which signifies the difficulty to cap-ture the corresponding natural frequencies with a desired level ofaccuracy. Moreover, Table 8 also provide ABAQUSmodels solutionsfor both reduced and full integration scheme in order to underlinethe numerical deficiencies in FE solutions. The core modesobtained via the 6� 9L16 model, with m = 1–5, are shown in
Fig. 14. Comparison of the fifth modes shape for m = 2–5, by 48L16 and 3D FEM model.
Y. Yang et al. / Composite Structures 175 (2017) 28–45 43
Fig. 8. Observing these mode shapes, we can see that core modesoccur accompanied by the significant deformation of the soft core,which are characterised by symmetric and anti-symmetric modeshapes in sequential order.
3.3. T-shaped composite beam
After assessing the performance of the LE method in compositebeams with rectangular cross sections, a T-shaped thin-walledcomposite beam is then considered (see Fig. 9). The structure has
the following geometric characteristics: width b ¼ 0:1 m, heighth ¼ 0:2 m, slenderness ratio l=b ¼ 10, thickness of flanget1 ¼ 0:01 m, thickness of web t2 ¼ 0:01 m. The flange is composedof two cross-ply laminations [0/90] of the same thickness, whilethe web is made up of one lamination [0]. The material propertiesare: EL ¼ 144 MPa, ET ¼ 9:65 MPa, GLT ¼ 4:14 MPa, GTT ¼ 3:45 MPa,mLT ¼ mLT ¼ 0:3, q ¼ 1389 kg=m3. Table 9 shows the first five natu-ral frequencies with m = 1–5 by the LE model and 3D FEM model.As may be noted from Table 9, the lower-order LE model (L4) pro-vides good results with enough DOFs and the higher-order LE mod-
44 Y. Yang et al. / Composite Structures 175 (2017) 28–45
els (L9 or L16) produce more accurate results than 3D FEM model.The lowest mode shapes corresponding to 1–9 via the 29L16model, with m = 1–4 are displayed in Fig. 10. As shown in Fig. 10,mode 1 is always featured by flexural mode on plane xy, whateverthe value of m. Mode 2 for m = 1 is torsion mode, being shell-likemode for other values of m. The flexural mode on plane yz tendsto appear after the aforementioned three modes.
3.4. Single-bay composite box beam
In this section, further study is performed for the case of asingle-bay composite box beam. The configuration of the cross sec-tion can be seen in Fig. 11. The dimensions of the cross section areb ¼ 0:1 m and h ¼ 0:2 m. The length to width ratio is l=b ¼ 10. Thethickness of the wall is t ¼ 0:01. As in the previous analysis case,two layers [0/90] are included in the top and bottom flange,respectively. One layer [0] is employed for two webs. An orthotro-pic material is adopted for each layer in conformity to the case of T-shaped cross section. Table 10 shows the first five non-dimensionalnatural frequencies with m = 1–5 acquired by the present modeland 3D FEM model. Compared with the L4 results in the case ofT-shaped and box beams, the proposed model is proved with poorcapability to capture the warping phenomena. Thus, a higher-ordermodel with enough DOFs is imperative in this case. In theend, comparison of the fifth mode shapes for m = 2–5, by 32L16and 3D FEM model are shown in Fig. 12, providing satisfactoryresults.
3.5. Composite sandwich-box beam
The final example wants to demonstrate the enhanced andunique capability of the proposed LW beam model to address 3Dproblems. The cross section of the composite structure consideredis thus shown in Fig. 13. The same geometrical shape of the crosssection as for previous case is account for again, including a two-layer [0/90] laminate in the top and bottom faces, respectively,one layer [0] in the left and right faces, respectively, and a soft coreis added in the middle. Also, the same material properties of theface and core are adopted as considered in the case of three-layercomposite beam. Results are reported in Table 11. Although thisstructure is more complicated than the single-bay composite boxbeam, all modes under consideration are capable of being detectedprecisely via present 50L9 and 48L16 models. In particular, thecomparison between 48L16 and 3D FEM model in forecasting thefifth mode shapes for m = 2–5 is presented in Fig. 14. It is obviousthat 48L16 model can describe the bending and core deformationphenomena correctly.
4. Conclusions
In this paper, a unified closed-form formulation of refined beammodels has been extended to the free vibration of simply sup-ported cross-ply composite beams. The analysis has been per-formed in the domain of Carrera Unified Formulation, where 3Dkinematic fields can be discretized as the expansion of any orderof the cross-sectional node displacement unknowns via LagrangeExpansion (LE), being the ability of layer-wise naturally satisfied.The strong-form governing equation, derived by the principle ofvirtual displacement, can be solved by a Navier-type closed-formsolution through the assumption of simply supported boundaryconditions. Several numerical cases have been carried out todemonstrate the accuracy and effectiveness of the proposedmethodology in comparison with 3D FEM results obtained fromcommercial code, including long and short cross-ply laminatebeams with different stacking sequences, thin-walled composite
beams and composite sandwich beams. From these results, the fol-lowing conclusions can be drawn:
1. LE CUF model are considered to yield similar results as 3D FEMresults, and more accurately than TE CUF model. This conclu-sion is more evident in the case of short compact beams.
2. Non-classical modes such as torsion, shear and axial/shear cou-pling modes can be detected with higher-order CUF LE model.Moreover, order of mode appearance may be interchanged eachother for higher half wave numbers as the number of layerincreases, which can be also captured by higher-order CUF LEmodel precisely.
3. In the case of heterogeneous structures with different materialproperties (e.g., sandwich beams) and when several natural fre-quencies fall in a narrow frequency spectrum, the use of lower/order beam models is not recommended.
4. Lower-order CUF LE model gives unsatisfactory mode results inthe case of thin-walled composite beams (e.g., T-shaped andsingle-bay box shape). Meanwhile, higher-order CUF LE modelwith enough DOFs is capable of capturing the shell-like modes.
5. Concerning the beams with complex material properties (e.g.,composite sandwich beams), the present model readily showsits high-efficiency over 3D FEM solutions.
Acknowledgments
The first author acknowledges the support by the scholarshipfrom the China Scholarship Council (CSC) (Grant No.201606710014) and Fundamental Research Funds for the CentralUniversities (Grant No. 2014B31414).
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